The concept of inverse systems for standard and positive linear systems is introduced. Necessary and sufficient conditions for the existence of the positive inverse system for continuous-time and discrete-time linear systems are established. It is shown that: 1) The inverse system of continuous-time linear system is asymptotically stable if and only if the standard system is asymptotically stable. 2) The inverse system of discrete-time linear system is asymptotically stable if and only if the standard system is unstable. 3) The inverse system of continuous-time and discrete-time linear systems are reachable if and only if the standard systems are reachable. The considerations are illustrated by numerical examples.
This paper presents the method of analysis of parametric systems in frequency domain. These systems are also referred to as linear time varying systems (LTV). The article includes a description of an analytical method for determining the frequency response of the first order parametric circuit with non-periodically variable parameters. The results have been illustrated by an example.
The stability of positive linear continuous-time and discrete-time systems is analyzed by the use of the decomposition of the state matrices into symmetrical and antisymmetrical parts. It is shown that: 1) The state Metzler matrix of positive continuous-time linear system is Hurwitz if and only if its symmetrical part is Hurwitz; 2) The state matrix of positive linear discrete-time system is Schur if and only if its symmetrical part is Hurwitz. These results are extended to inverse matrices of the state matrices of the positive linear systems.
Necessary and sufficient conditions for robust stability of the positive discrete-time interval system with time-delays are established. It is shown that this system is robustly stable if and only if one well de?ned positive discrete-time system with time-delays is asymptotically stable. The considerations are illustrated by numerical example.
A new method for computation of positive realizations of given transfer matrices of fractional linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by examples.
A new concept (notion) of the practical stability of positive fractional discrete-time linear systems is introduced. Necessary and sufficient conditions for the practical stability of the positive fractional systems are established. It is shown that the positive fractional systems are practically unstable if corresponding standard positive fractional systems are asymptotically unstable.
New tests (criterions) for checking the reachability and the observability of positive linear-discrete-time systems are proposed. The tests do not need checking of rank conditions of the reachability and observability matrices of the systems. Simple sufficient conditions for the unreachability and unobservability of the systems are also established.
A new class of positive fractional 2D hybrid linear systems is introduced. The solution of the hybrid system is derived. The classical Cayley-Hamilton theorem is extended for fractional 2D hybrid systems. Necessary and sufficient conditions for the positivity are established.
New frequency domain methods for stability analysis of linear continuous-time fractional order systems with delays of the retarded type are given. The methods are obtained by generalisation to the class of fractional order systems with delays of the Mikhailov stability criterion and the modified Mikhailov stability criterion known from the theory of natural order systems without and with delays. The study is illustrated by numerical examples of time-delay systems of commensurate and non-commensurate fractional orders.
The concept of strong stability is extended for positive and compartmental linear systems. It is shown that: 1) the asymptotically stable positive and compartmental systems are strongly stable if the eigenvalues of the system matrix are distinct, 2) electrical circuits consisting of resistances, capacitances (inductances) and source voltages are strongly stable.
The positivity of fractional descriptor linear continuous-time systems is investigated. The solution to the state equation of the systems is derived. Necessary and sufficient conditions for the positivity of fractional descriptor linear continuous-time systems are established. The considerations are illustrated by numerical examples.
Necessary and sufficient conditions for the reachability and observability of the positive electrical circuits composed of resistors, coils, condensators and voltage sources are established. Definitions of the input-decoupling zeros, output-decoupling zeros and input-output decoupling zeros of the positive electrical circuits are proposed. Some properties of the decoupling zeros of positive electrical circuits are discussed.
In this paper, the results of correlations between air temperature and electricity demand by linear regression and Wavelet Coherence (WTC) approach for three different European countries are presented. The results show a very close relationship between air temperature and electricity demand for the selected power systems, however, the WTC approach presents interesting dynamics of correlations between air temperature and electricity demand at different time-frequency space and provide useful information for a more complete understanding of the related consumption.
Simple new necessary and sufficient conditions for asymptotic stability of the positive linear discrete-time systems with delays in states are established. It is shown that asymptotic stability of the system is equivalent to asymptotic stability of the corresponding positive discrete-time system without delays of the same size. The considerations are illustrated by numerical examples.
Given a linear discrete system with initial state x0 and output function yi , we investigate a low dimensional linear systemthat produces, with a tolerance index ǫ, the same output function when the initial state belongs to a specified set, called ǫ-admissible set, that we characterize by a finite number of inequalities. We also give an algorithm which allows us to determine an ǫ-admissible set.
Simple necessary and sufficient conditions for robust stability of the positive linear discrete-time systems with delays with linear uncertainty structure in two cases: 1) unity rank uncertainty structure, 2) non-negative perturbation matrices, are established. The proposed conditions are compared with the suitable conditions for the standard systems. The considerations are illustrated by numerical examples.
This paper considers the feasibility of different technologies for an electromagnetic launcher to assist civil aircraft take-off. This method is investigated to reduce the power required from the engines during initial acceleration. Assisted launch has the potential of reducing the required runway length, reducing noise near airports and improving overall aircraft efficiency through reducing engine thrust requirements. The research compares two possible linear motor topologies which may be efficaciously used for this application. The comparison is made on results from both analytical and finite element analysis (FEA).
The minimum energy control problem for the positive descriptor discrete-time linear systems with bounded inputs by the use of Weierstrass-Kronecker decomposition is formulated and solved. Necessary and sufficient conditions for the positivity and reachability of descriptor discrete-time linear systems are given. Conditions for the existence of solution and procedure for computation of optimal input and the minimal value of the performance index is proposed and illustrated by a numerical example.
This paper presents the results of the DFG-project (Deutsche Forschungsgemeinschaft) Q-ELF (“Qualitätsorientierter Methodenworkflow für die Produktneuentwicklung eines Linearantriebs in der Fördertechnik”) carried out in cooperation of the TU Dortmund University (support code KU 1307/12-1) with the BUW Wuppertal (support code WI 1234-11/1). The project continues the former project SFB 696 (Sonderforschungsbereich) regarding the Demand Compliant Design (DeCoDe) and the corresponding system model that strengthens the knowledge management to create high-quality mechatronical systems. In contrast to the SFB, which comprised the reverse engineering of a belt conveyor, Q-ELF applied a workflow of methods for quality oriented development on a new product. The DeCoDe ensures a methodical development that connects different engineering domains. This connection is important because the most problems and malfunctions arise at the interface of different domains due to their different notations for example. This approach also enables a methodical comparison of different competing concepts to pick the best suited one. A genetic algorithm is presented to further decrease the design-space. The project was carried out to develop linear drives for intralogistic systems.
In intra-enterprise logistics and automation of manufacturing processes general a rising productivity by high flexibility is required. Existing transportation systems exclusively use two-dimensional track sections, because they can be served with standard drives. Because of these simple structures the transport speed is limited and thereby also the throughput. In this paper now a modular transportation system is presented which could reach higher speeds with a direct drive and the use of centrifugal force compensating curves. Simultaneously the system also can change the altitude. All this succeeds with the integration of three-dimensional track sections. Therefore a two piped guiding system with a long stator linear motor was designed. To combine the linear motor with the three dimensional track special stator elements were developed which allow a bending of the stator to follow the route course. The current work deals with the implementation of a mechanical passive switch, which is operated by the electromagnetic forces of the linear motor. So no additional mechanical actors or a separate electromagnetic system are necessary.
This paper presents an analytical approach for solving the weighting matrices selection problem of a linear quadratic regulator (LQR) for the trajectory tracking application of a magnetic levitation system. One of the challenging problems in the design of LQR for tracking applications is the choice of Q and R matrices. Conventionally, the weights of a LQR controller are chosen based on a trial and error approach to determine the optimum state feedback controller gains. However, it is often time consuming and tedious to tune the controller gains via a trial and error method. To address this problem, by utilizing the relation between the algebraic Riccati equation (ARE) and the Lagrangian optimization principle, an analytical methodology for selecting the elements of Q and R matrices has been formulated. The novelty of the methodology is the emphasis on the synthesis of time domain design specifications for the formulation of the cost function of LQR, which directly translates the system requirement into a cost function so that the optimal performance can be obtained via a systematic approach. The efficacy of the proposed methodology is tested on the benchmark Quanser magnetic levitation system and a detailed simulation and experimental results are presented. Experimental results prove that the proposed methodology not only provides a systematic way of selecting the weighting matrices but also significantly improves the tracking performance of the system.